Annals of Applied Probability

Randomly growing braid on three strands and the manta ray

Jean Mairesse and Frédéric Mathéus

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Abstract

Consider the braid group B3=〈a, b|aba=bab〉 and the nearest neighbor random walk defined by a probability ν with support {a, a−1, b, b−1}. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on B3 quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 2 (2007), 502-536.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1174323255

Digital Object Identifier
doi:10.1214/105051606000000754

Mathematical Reviews number (MathSciNet)
MR2308334

Zentralblatt MATH identifier
1146.60009

Subjects
Primary: 20F36: Braid groups; Artin groups 20F69: Asymptotic properties of groups 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy)

Keywords
Braid group B_3 random walk drift harmonic measure dihedral Artin group

Citation

Mairesse, Jean; Mathéus, Frédéric. Randomly growing braid on three strands and the manta ray. Ann. Appl. Probab. 17 (2007), no. 2, 502--536. doi:10.1214/105051606000000754. https://projecteuclid.org/euclid.aoap/1174323255


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